0580/43 May/Jun 2024, Question 5: Solving Equations by Graphical Methods

Part (ai) · 1 mark

Read the value of f(2) from the graph.

1. 1Go up from x = 2 to the curve and across: f(2) = 3

Answer3

Part (aii) · 3 marks

Use the graph to find all x values where f(x) = 5.

1. 1Draw the line y = 5 across the grid
2. 2It meets the left branch at x = −2 and x = −0.42
3. 3It meets the right branch at x = 2.42

Answer−2, −0.45 to −0.4, 2.40 to 2.45

Part (aiii) · 1 mark

State the value of k for which f(x) = k has exactly two solutions.

1. 1A horizontal line cutting the curve exactly twice must touch the minimum of the left branch
2. 2The minimum point is at y = 3, so k = 3

Answer3

Part (aiv) · 1 mark

Choose from tangent, asymptote, root, perpendicular to describe the line x = 0 for y = f(x).

1. 1The curve gets closer and closer to x = 0 but never meets it
2. 2x = 0 is an asymptote

Answerasymptote

Part (bi) · 2 marks

Draw the straight line y = x − 2 on the grid for x from −3 to 3.

1. 1Points: (−3, −5), (0, −2), (3, 1)
2. 2Plot them and join with a ruled straight line

AnswerCorrect ruled line y = x − 2 from (−3, −5) to (3, 1)

Part (bii) · 1 mark

Find the x value where f(x) equals x − 2.

1. 1The line y = x − 2 crosses the curve at x = 1

Answer1

Part (c) · 2 marks

Using f(x) = x² − c/x and the graph, show that c = 2.

1. 1(−1, 3) is on the graph
2. 23 = (−1)² − c/(−1)
3. 33 = 1 + c
4. 4c = 2

AnswerSubstituting a point on the graph, e.g. (−1, 3): 3 = (−1)² − c/(−1) = 1 + c, so c = 2

Part (d) · 2 marks

Rewrite f(x) = x − 2 as x³ + px² + qx = 2 and find p and q.

1. 1x² − 2/x = x − 2
2. 2Multiply by x: x³ − 2 = x² − 2x
3. 3x³ − x² + 2x = 2
4. 4p = −1, q = 2

Answerp = −1, q = 2

More past-paper questions on solving equations by graphical methods

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